Theorem 19.26 1479

Description: Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

Ref	Expression
19.26	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.26

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	alimi 1453	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜑)
3		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	alimi 1453	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜓)
5	2, 4	jca 306	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓))
6		id 19	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓))
7	6	alanimi 1457	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
8	5, 7	impbii 126	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.26-2  1480  19.26-3an  1481  albiim  1485  2albiim  1486  hband  1487  hban  1545  19.27h  1558  19.27  1559  19.28h  1560  19.28  1561  nford  1565  nfand  1566  equsexd  1727  equveli  1757  sbanv  1887  2eu4  2117  bm1.1  2160  r19.26m  2606  unss  3307  ralunb  3314  ssin  3355  intun  3871  intpr  3872  eqrelrel  4721  relop  4770  eqoprab2b  5923  dfer2  6526  omniwomnimkv  7155